In short: Most students know the formulas but freeze when they see a question. The problem isn't memorization — it's recognition. This guide maps question keywords to the exact formula you need. When the question says "find the ratio in which point P divides the line" → Section Formula. When it says "find the nature of roots" → Discriminant. No more staring at a question wondering which formula applies. This is your decision-tree cheat sheet for the CBSE Class 10 Maths 2027 board exam.
How This Guide Works
Every CBSE board question contains trigger words. These words tell you which formula to use before you even read the numbers. We've analyzed CBSE Class 10 Maths SQP 2027 and past papers to map every common question type to its formula.
The format is: "If the question says..." → Use this formula
Chapter 1: Real Numbers
| Question Trigger | Formula / Method | Example |
|---|---|---|
| "Find HCF / GCD of two numbers" | Prime factorization → HCF = product of smallest powers of common primes | HCF(12, 18) = 2¹ × 3¹ = 6 |
| "Find LCM of two numbers" | LCM = product of greatest powers of all primes | LCM(12, 18) = 2² × 3² = 36 |
| "Verify HCF × LCM = product of numbers" | HCF(a,b) × LCM(a,b) = a × b | 6 × 36 = 12 × 18 = 216 ✓ |
| "Prove √2 is irrational" | Contradiction method: assume p/q in lowest terms, square both sides, show contradiction | Standard 3-mark proof |
| "After how many decimal places will the decimal expansion terminate?" | If denominator in simplest form = 2ᵐ × 5ⁿ → terminates after max(m,n) places | 7/80 = 7/(2⁴×5¹) → 4 decimal places |
Chapter 2: Polynomials
| Question Trigger | Formula / Method |
|---|---|
| "Find the zeroes of the polynomial" | Set p(x) = 0, factorize, solve |
| "Find a quadratic polynomial whose zeroes are α and β" | p(x) = x² − (α + β)x + αβ |
| "Sum of zeroes" | α + β = −b/a (for ax² + bx + c) |
| "Product of zeroes" | αβ = c/a |
| "If one zero is given, find the other" | Use αβ = c/a → other zero = (c/a) ÷ known zero |
| "Verify relationship between zeroes and coefficients" | Check: α + β = −b/a AND αβ = c/a |
| "Find zeroes of a cubic polynomial" (3 zeroes) | α + β + γ = −b/a; αβ + βγ + αγ = c/a; αβγ = −d/a |
Chapter 3: Pair of Linear Equations
| Question Trigger | Formula / Method |
|---|---|
| "Solve the pair of equations" (graph method asked) | Plot both lines, intersection point is the solution |
| "Solve by substitution / elimination / cross-multiplication" | Use the specified method |
| "Find whether equations are consistent / inconsistent" | Compare a₁/a₂, b₁/b₂, c₁/c₂ |
| "Unique solution" | a₁/a₂ ≠ b₁/b₂ → intersecting lines |
| "No solution (inconsistent)" | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel lines |
| "Infinitely many solutions (dependent)" | a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident lines |
| "For what value of k will the system have no solution / unique solution / infinite solutions?" | Set up the ratios and solve for k |
Decision tree:
- See "consistent" → check if lines intersect or coincide
- See "inconsistent" → verify parallel condition
- See "value of k" → set up ratio condition and solve
Chapter 4: Quadratic Equations
| Question Trigger | Formula / Method |
|---|---|
| "Find the roots of the quadratic equation" | Quadratic formula: x = [−b ± √(b²−4ac)] / 2a |
| "Solve by factorisation" | Split middle term → equate each factor to 0 |
| "Find the nature of roots" | Discriminant: D = b² − 4ac |
| "Real and equal roots" | D = 0 |
| "Real and distinct roots" | D > 0 |
| "No real roots" | D < 0 |
| "For what value of k does the equation have equal roots?" | Set D = 0 → b² − 4ac = 0 → solve for k |
| "Form a quadratic equation whose roots are..." | x² − (sum of roots)x + (product of roots) = 0 |
This is the most tested chapter. The CBSE Class 10 Maths weightage shows Quadratic Equations carry 5–7 marks.
Chapter 5: Arithmetic Progressions
| Question Trigger | Formula / Method |
|---|---|
| "Find the nth term" | aₙ = a + (n−1)d |
| "Find the sum of first n terms" | Sₙ = n/2 [2a + (n−1)d] OR Sₙ = n/2 [a + l] (if last term l is given) |
| "Which term of the AP is...?" | Set aₙ = given value, solve for n |
| "How many terms are in the AP?" | Use aₙ = a + (n−1)d with last term |
| "Find the common difference" | d = a₂ − a₁ = a₃ − a₂ |
| "Is 0 a term of this AP?" | Set aₙ = 0, check if n is a positive integer |
| "Find 3 numbers in AP whose sum is..." | Let them be a−d, a, a+d (sum = 3a) |
| "Find 4 numbers in AP whose sum is..." | Let them be a−3d, a−d, a+d, a+3d (sum = 4a) |
| "Sum of first n terms is given as Sₙ = ... Find the AP" | aₙ = Sₙ − Sₙ₋₁ |
Practice CBSE Maths on Padhaao
AI tutor with 10 learning modes · Quiz, Flashcards, Exam Practice & more · 3 free chapters daily
Chapter 6: Triangles (Similarity)
| Question Trigger | Formula / Method |
|---|---|
| "Prove triangles are similar" | Use AA (two angles equal), SAS (one angle equal + sides proportional), SSS (all sides proportional) |
| "Find the unknown side in similar triangles" | Corresponding sides are proportional: AB/PQ = BC/QR = AC/PR |
| "Ratio of areas of two similar triangles" | (Area₁/Area₂) = (side₁/side₂)² |
| BPT: "Line parallel to one side divides the other two proportionally" | AD/DB = AE/EC (Basic Proportionality Theorem / Thales) |
| "Converse of BPT" | If AD/DB = AE/EC, then DE ∥ BC |
| "Pythagoras theorem" | In right △, hypotenuse² = base² + perpendicular² |
Chapter 7: Coordinate Geometry
| Question Trigger | Formula / Method |
|---|---|
| "Find the distance between two points" | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| "Show that points form a right triangle" | Find all 3 distances, check if sum of squares of two = square of third |
| "Show points are collinear" | Distance AB + BC = AC OR area of triangle = 0 |
| "Find the midpoint" | M = [(x₁+x₂)/2, (y₁+y₂)/2] |
| "Find the ratio in which a point divides the line" → | Section Formula: P = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)] |
| "Find the point that divides AB in ratio m:n" | Section formula (above) |
| "Find the area of triangle given three vertices" | **Area = ½ |
| "Find the centroid" | G = [(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3] |
The section formula trap: When the question says "in what ratio does the x-axis divide..." → set y-coordinate = 0 in section formula and solve for m:n. For y-axis → set x = 0.
Chapter 8: Introduction to Trigonometry
| Question Trigger | Formula / Method |
|---|---|
| "Find the value of sin/cos/tan for given angle" | Use standard table: 0°, 30°, 45°, 60°, 90° |
| "Prove the identity" | Start from LHS, manipulate using: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ |
| "If sinA = 3/5, find cosA and tanA" | Draw right triangle OR use sin²A + cos²A = 1 |
| "Evaluate: sin²30° + cos²60° + ..." | Substitute standard values and compute |
Standard Values Table:
| θ | sinθ | cosθ | tanθ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Chapter 9: Some Applications of Trigonometry
| Question Trigger | Formula / Method |
|---|---|
| "Angle of elevation" | Object is ABOVE eye level → look UP |
| "Angle of depression" | Object is BELOW eye level → look DOWN (angle = angle of elevation from the lower point) |
| "Find the height of tower/building" | tanθ = height/base → height = base × tanθ |
| "Find the distance from the foot" | base = height/tanθ = height × cotθ |
| "Two angles of elevation from two points" | Set up two equations with tanθ₁ and tanθ₂, eliminate common variable |
| "Shadow problems" | Shadow length = base, height of object = perpendicular |
Key rule: Always draw the right triangle first. Label: vertical = height (what you find), horizontal = distance, angle at ground = elevation. The board exam time management guide recommends drawing diagrams for every trig question — it takes 30 seconds but prevents wrong setups.
Chapter 10: Circles
| Question Trigger | Formula / Method |
|---|---|
| "Tangent from an external point" | Two tangents from the same point are equal: PA = PB |
| "Prove tangent is perpendicular to radius" | Standard theorem: OA ⊥ tangent at point of contact |
| "Find the length of tangent from a point to a circle" | Length = √(d² − r²) where d = distance from centre to external point |
Chapter 11: Areas Related to Circles
| Question Trigger | Formula / Method |
|---|---|
| "Find area of circle" | A = πr² |
| "Find circumference" | C = 2πr |
| "Area of sector" | A = (θ/360) × πr² |
| "Length of arc" | L = (θ/360) × 2πr |
| "Area of segment" | Area of sector − Area of triangle |
| "Area of ring / shaded region between two circles" | π(R² − r²) |
Chapter 12: Surface Areas and Volumes
| Question Trigger | Formula / Method |
|---|---|
| "Cone on top of cylinder" | Total SA = CSA of cylinder + CSA of cone + area of base |
| "Hemisphere on cylinder" | Total SA = CSA of cylinder + CSA of hemisphere + area of bottom base |
| "Metallic sphere melted and recast into smaller spheres" | Volume conserved: (4/3)πR³ = n × (4/3)πr³ → n = (R/r)³ |
| "Cone + hemisphere toy" | Volume = (1/3)πr²h + (2/3)πr³ |
| "Frustum of a cone" | V = (πh/3)(r₁² + r₂² + r₁r₂) |
| "Frustum slant height" | l = √[h² + (r₁ − r₂)²] |
| "Frustum CSA" | π(r₁ + r₂)l |
| "Water flowing through pipe into tank" | Volume of water per second = area of pipe × speed; time = tank volume / flow rate |
Chapter 13: Statistics
| Question Trigger | Formula / Method |
|---|---|
| "Find the mean" (grouped data) | Direct: x̄ = Σfᵢxᵢ / Σfᵢ |
| "Find the mean" (large values) | Assumed mean: x̄ = a + Σfᵢdᵢ/Σfᵢ (where dᵢ = xᵢ − a) |
| "Find the mean" (step deviation) | x̄ = a + h × (Σfᵢuᵢ/Σfᵢ) (where uᵢ = dᵢ/h) |
| "Find the median" | Median = l + [(n/2 − cf)/f] × h |
| "Find the mode" | Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h |
| "Less than / More than ogive" | Plot cumulative frequency → median at intersection of both ogives |
Where: l = lower boundary of median/modal class, h = class width, f = frequency of median/modal class, cf = cumulative frequency before median class, n = total frequency, f₀ = frequency before modal class, f₁ = modal class frequency, f₂ = frequency after modal class.
Chapter 14: Probability
| Question Trigger | Formula / Method |
|---|---|
| "Find the probability of event E" | P(E) = favourable outcomes / total outcomes |
| "Probability of NOT happening" | P(not E) = 1 − P(E) |
| "Two dice thrown" | Total outcomes = 36, list and count favourable |
| "Deck of cards" | Total = 52; Hearts/Diamonds/Clubs/Spades = 13 each; Face cards = 12; Aces = 4 |
| "Bag of balls" — drawing problems | Count favourable colour / total balls |
The Master Decision Tree
When you open a question paper, follow this:
- Read the question trigger words (find, prove, show, for what value of k, etc.)
- Identify the chapter from the topic
- Match trigger words to formula using the tables above
- Write the formula first, then substitute values
- Check units and sign — especially in coordinate geometry (negative coordinates are common traps)
This approach works with the how to study smart not hard methodology — pattern recognition beats rote memorization.
Frequently Asked Questions
How is this different from a formula list?
A formula list tells you WHAT the formula is. This guide tells you WHEN to use it. Most students memorize Section Formula but don't recognize "in what ratio does the y-axis divide the line joining..." as a Section Formula question. This guide bridges that gap with trigger-word mapping.
Which chapters have the most formula-based questions in the board exam?
Coordinate Geometry, Trigonometry (Applications), Statistics, and Surface Areas & Volumes are the most formula-heavy chapters. Together they carry about 25–30 marks of formula-application questions. See the complete CBSE Class 10 Maths chapter-wise weightage for exact numbers.
What if a question doesn't match any trigger word?
If you cannot identify the trigger, look at what information is given (coordinates, angles, radii, frequencies) and what is asked. The given data type usually points to the chapter. Coordinates → Coordinate Geometry. Frequencies → Statistics. Angles with heights → Trigonometry applications.
Should I memorize all these formulas?
Yes, but through practice, not rote memorization. Solve 5 problems per formula type. After the third problem, you will automatically associate the question trigger with the formula. By the fifth, it becomes automatic. Use previous year papers for timed practice.
How do I handle questions that need two formulas combined?
Some questions require chaining formulas. For example: "Find the area of a triangle formed by mid-points of a given triangle" needs midpoint formula (3 times) + area of triangle formula. Read the question fully, identify both formulas, solve step by step. These carry 4–5 marks and are where toppers pull ahead.
Is the formula application the same for the second board exam attempt?
Yes. The CBSE two-exam system uses the same syllabus and formula set for both attempts. Question difficulty may vary slightly but the formula-to-question mapping remains identical. See the CBSE two exam system guide for details.
Related Guides
- CBSE Class 10 Maths Chapter Wise Marks Weightage 2027
- CBSE Class 10 Maths Formulas Complete List
- CBSE Class 10 Maths 30 Most Expected Questions 2027
- Board Exam Answer Sheet — Write for Maximum Marks 2027
- How to Study Smart Not Hard — Indian Students
- Board Exam Time Management — 3 Hour Strategy 2027
Practice CBSE Class 10 Maths Formula Application on Padhaao →
₹21/mo (annual plan)
Padhaao covers every CBSE Maths chapter
1,005 chapters across all subjects. 2.5 crore+ students. Pro from ₹21/month (annual plan).
Practice CBSE Maths chapters with AI
Every chapter from your syllabus. 10 learning modes. 3 free every day.
Start Maths